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Let (81,82) be solution of the system (0uXi 012X2 (21X1 (22*2 = 0 Show that (cSt cSz) is also solution for any real number €.Show that the linear system(114[...

Question

Let (81,82) be solution of the system (0uXi 012X2 (21X1 (22*2 = 0 Show that (cSt cSz) is also solution for any real number €.Show that the linear system(114[ 413X2 013+3 (az1*1 421*2 02343 = cannot have unique solution_

Let (81,82) be solution of the system (0uXi 012X2 (21X1 (22*2 = 0 Show that (cSt cSz) is also solution for any real number €. Show that the linear system (114[ 413X2 013+3 (az1*1 421*2 02343 = cannot have unique solution_



Answers

Check your solution in Exercise 109 , showing that it satisfies all three equations of the system.

So reading his linear equations is pretty straight forward. We're gonna have X plus five Z because negative nine from our first round in the Matrix. Then we'll have why minus two Z equals four from our second row and then where I am, zero z equals three from Mark their growth. Now, from this third equation, we can see that zero equals three. So, in fact, this linear system will never be satisfied. Because this is a contradiction on this can never happen, so there is no solution.

Here we have the matrix form of a system of of equations, and we're going to change it back into the equation form. So from the top row, using the 10 and the negative three as the coefficients on X y and Z, we end up with X plus zero y minus three Z equals seven and from row two using the 01 and two as the coefficients on X, y and Z, we have zero X, we have one. Why? And we have to Z equals negative five. And from row three it's just zero equals zero. So that doesn't give us anything. Will there be a unique solution to this system? No, there will not. This has infinitely many solutions. The best we can do is just express x and y in terms of Z. So because say, that X will be seven plus three z and we could choose whatever number we wanted for Z and find X based on that number. We could do something like that for why and say that why is equal to negative five minus two Z. So, for example, if we picked zero for Z, that would give us uh, an accident. Why? Value? If we picked one for Z, we would get another X and y value, so we've infinitely many points.

Alright. The first thing you want to do in solving this system of equations is find the determinant of the matrix made up all the coefficients, the F K one and one. And then for the wise you have one K one and disease you have 11 in K. And you're gonna leave this um The constant sound with this one. I just found the determinant first. All right, so you should get change the third plus one plus one minus k minus K as K. And it should simplify into K two, third minus three K plus two. And then you can factor this, yep, you're going back to this into hold him Okay minus one squared K plus two. Okay. And so what you do this you're looking for first per eight is a unique solution and you will always have a unique solution if the determinant does not equal zero. Alright, so all you have to do is say the determinant, which we just found and as long as this does not equal zero. So you can break these apart and say we have an unique solution. As long as came out to one square just and equal zero. And keep list too. Yes, I mean Uh huh. Moment. Yeah. All right. So as long as you sort this out and you get longest kate is not equal one and K does not equal negative to you have a unique solution. So now you're looking for infinitely many and no solution. So these are the two numbers that are going to give you either or so be that is the B. B is going to be see infinitely many. So in order to have impudently many solutions your matrix, no, no matter what you have appear, the bottom should be 000 All zeros. Okay, so we have look Okay. 111 Okay. One We're gonna put him into an augmented matrix using the constants of 11 and one in her original problem. So we want to make this look like you know we want to make it to where it's all zeros in the bottom. So we can start first. We can do um wrote you minus row three. That is a three to give us a new road to. And then we can also do the same thing to give us our row three because we're trying to get a zero what? In the in the bottom? Okay so first we're gonna start with the beginning will say the top ones will stay same mm. Okay two all right lets you know what actually let's do you write three first and see what happens? We got row two months bro three. So yeah one minutes one is we're sleeping for now. You got one minute 10 next when you have one minus K and then k minus one. My keys are coming out too good today and then you have one minus one which will give you zero. So if we wanted our entire bottom losing believe this went out for now because we can see it already. If we wanted our entire bottom road we all zeros. That means these two would have to equal zero as well. Right? So initially there is the same thing. So after equal zero and the k minus one, I would have to equal zero. And they both tell us that okay when K equals one we'll have all zeros on the bottom two years ago as your of our matrix augmented matrix. And then that would tell us that we have infinitely many solutions. So when K equals one we will have infinitely many solutions. And that is many infinitely many solutions. Okay. And so for C we're looking for that is to see no solution. What's going on my pen? But the only other option we have is the K equals negative two. So when K equals negative two, this is sorry, that's not very good, no solution. Um when kate was negative too. So whenever you factor and you figure out, you know, your original she numbers one will be in Flamini, one will be your next solution and that's it. Good job. Thank you.

We'll start by multiplying the top equation by two. So on the top will have no 20.2 x with why is equal to one point for on the bottom or stay the same. So we'll have no 0.2 x first, no 0.7. Why is equal to no 0.8? So we're going to subtract the bottom from the top. This will give us a 0.3. Why is equal to 0.6? Which gives us why equals two. Substituting this back into the original. We have 0.2 x It was too equals. One point for this gives those there a point to acts is equal to minus 0 00.6 or X is equal Tu minus three.


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